11.1 Sub-Riemannian manifolds and Carnot groups
4 min read•august 14, 2024
Sub-Riemannian manifolds blend smooth geometry with restricted motion, creating unique spaces where distance is measured along special paths. These structures challenge our usual notions of dimension and distance, offering a fresh perspective on geometric relationships.
Carnot groups are key players in this field, serving as local models for sub-Riemannian spaces. They help us understand the intricate geometry of these manifolds, much like how flat spaces help us grasp curved surfaces in everyday life.
Sub-Riemannian Manifolds
Definition and Key Properties
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- A is a smooth manifold M equipped with a smooth distribution H ⊂ TM of subspaces of the tangent spaces, called a , and a smoothly varying inner product g defined only on H
- The dimension of the horizontal distribution H is strictly less than the dimension of the manifold M
- The horizontal distribution H is required to satisfy the
- States that the horizontal vector fields, along with their Lie brackets, should span the entire tangent space at each point of the manifold
- The distance between two points on a sub-Riemannian manifold is defined as the infimum of the lengths of all connecting the points
- The length is measured using the sub-Riemannian metric g
- in sub-Riemannian geometry are horizontal curves that locally minimize the sub-Riemannian distance between points
Hausdorff Dimension
- The of a sub-Riemannian manifold is typically greater than its topological dimension
- This is due to the presence of the horizontal distribution and the Hörmander condition
- Example: The , a basic example of a sub-Riemannian manifold, has topological dimension 3 but Hausdorff dimension 4
- The increased Hausdorff dimension reflects the intricate geometry and the presence of non-smooth horizontal curves in sub-Riemannian manifolds
Horizontal Distributions in Geometry
Definition and Role
- A horizontal distribution H on a smooth manifold M is a smooth assignment of a subspace Hp ⊂ TpM of the tangent space at each point p ∈ M
- The horizontal distribution determines the set of admissible curves, called horizontal curves, along which motion is allowed in sub-Riemannian geometry
- A curve γ : [0, 1] → M is said to be horizontal if its tangent vector γ'(t) belongs to the horizontal distribution Hγ(t) for almost every t ∈ [0, 1]
- The choice of the horizontal distribution significantly influences the geometry and properties of the sub-Riemannian manifold
- Affects the distance function, geodesics, and the Hausdorff dimension
Hörmander Condition
- The Hörmander condition requires the horizontal vector fields and their Lie brackets to span the entire tangent space at each point
- Ensures that any two points on the manifold can be connected by a horizontal curve
- Crucial for the controllability and connectivity of the sub-Riemannian manifold
- Example: In the Heisenberg group, the horizontal distribution is spanned by two vector fields X and Y, and their Lie bracket [X, Y] spans the vertical direction, satisfying the Hörmander condition
Carnot Groups and Sub-Riemannian Manifolds
Definition and Properties
- A G is a connected, simply connected whose 𝔤 admits a
- Stratification: a direct sum decomposition 𝔤 = V1 ⊕ V2 ⊕ ... ⊕ Vr such that [V1, Vj] = Vj+1 for 1 ≤ j < r and [V1, Vr] = {0}
- The subspace V1 is called the , and it generates the entire Lie algebra 𝔤 through the Lie bracket operation
- A Carnot group G can be equipped with a natural sub-Riemannian structure
- Considers the left-invariant distribution H corresponding to the horizontal layer V1 and a left-invariant metric g on H
Relationship to Sub-Riemannian Manifolds
- Carnot groups serve as local models for sub-Riemannian manifolds
- Similar to how Euclidean spaces serve as local models for Riemannian manifolds
- The at almost every point of a sub-Riemannian manifold is isometric to a Carnot group
- Known as the
- Carnot groups provide a rich class of examples of sub-Riemannian manifolds with a high degree of symmetry and a well-understood structure
- Example: The Heisenberg group is the simplest non-trivial example of a Carnot group and plays a fundamental role in sub-Riemannian geometry
Riemannian vs Sub-Riemannian Geometries
Metric Structure
- Riemannian geometry: based on a smoothly varying inner product (Riemannian metric) defined on the entire tangent space at each point of the manifold
- Sub-Riemannian geometry: relies on a smoothly varying inner product defined only on a subspace (horizontal distribution) of the tangent space
Distance and Geodesics
- Riemannian geometry: distance between two points is the infimum of the lengths of all smooth curves connecting them
- Geodesics are curves that locally minimize the Riemannian distance and are characterized by the vanishing of their covariant derivative
- Sub-Riemannian geometry: distance is the infimum of the lengths of only the horizontal curves connecting the points
- Geodesics are horizontal curves that locally minimize the sub-Riemannian distance and are characterized by the Pontryagin maximum principle
Dimension and Curvature
- Riemannian manifolds: Hausdorff dimension coincides with its topological dimension
- Well-developed theory of curvature plays a crucial role in understanding the geometry and topology
- Sub-Riemannian manifolds: Hausdorff dimension is typically greater than its topological dimension
- Notion of curvature is more intricate and is an active area of research
Classical Results
- Many classical results in Riemannian geometry, such as the Hopf-Rinow theorem and the Bishop-Gromov volume comparison theorem, do not hold or require significant modifications in the sub-Riemannian setting
- Example: The Hopf-Rinow theorem, which guarantees the completeness of Riemannian manifolds under certain conditions, does not hold in general for sub-Riemannian manifolds
Key Terms to Review (24)
Carnot Group: A Carnot group is a type of mathematical structure that represents a specific class of nilpotent Lie groups, which are equipped with a family of vector fields that satisfy certain conditions. These groups are fundamental in the study of sub-Riemannian geometry and are characterized by their stratified structure, where the group's Lie algebra can be decomposed into layers or levels. This structure plays a key role in understanding the geometry of spaces with non-Euclidean metrics.
Élie Cartan: Élie Cartan was a French mathematician known for his significant contributions to differential geometry, group theory, and the study of Riemannian and sub-Riemannian geometries. His work laid the foundation for understanding the geometric structures underlying various mathematical fields, including the study of Carnot groups, which are a class of nilpotent Lie groups that serve as models for sub-Riemannian manifolds.
Geodesics: Geodesics are the curves that represent the shortest path between two points on a surface or in a given space, often described in the context of Riemannian and sub-Riemannian geometries. They generalize the concept of straight lines to curved spaces, providing critical insights into the geometry and topology of various manifolds. Understanding geodesics helps in analyzing distances, minimizing paths, and exploring the intrinsic geometric structure of spaces, particularly in more complex settings like sub-Riemannian manifolds and variational problems involving mean curvature.
Gromov-Hausdorff Tangent Space: The Gromov-Hausdorff tangent space is a concept that extends the idea of tangent spaces in differential geometry to the context of Gromov-Hausdorff convergence, which is used to analyze the asymptotic behavior of metric spaces. This notion is particularly relevant when studying sub-Riemannian manifolds and Carnot groups, where the geometric structure is not necessarily smooth, and traditional notions of differentiation may not apply. The Gromov-Hausdorff tangent space captures the local geometric properties of these spaces at various scales, allowing for a deeper understanding of their structure.
Growth Condition: A growth condition refers to a requirement on the rate at which certain functions, typically distance functions or measures, can grow in relation to the structure of the space, especially in contexts involving sub-Riemannian manifolds and Carnot groups. It often ensures that geometric and analytical properties behave predictably under various transformations and can impact how distances are measured and understood within these spaces.
Hausdorff Dimension: Hausdorff dimension is a concept that extends the notion of dimensionality beyond integers, providing a way to measure sets that are too 'irregular' to fit traditional dimensions. It captures the complexity of a set's structure and is particularly useful for analyzing fractals, which often exhibit non-integer dimensions. This measurement plays a crucial role in understanding the relationship between various types of measures and helps in the study of geometric properties in more abstract settings.
Heisenberg group: The Heisenberg group is a mathematical structure that arises in the study of non-commutative geometry, particularly in sub-Riemannian geometry and control theory. It can be represented as a group of upper triangular matrices, which captures the essence of quantum mechanics and is fundamental in understanding certain geometric properties of spaces that involve curvature. This group has significant implications in areas such as robotics and control theory due to its unique properties related to motion and constraints.
Horizontal curves: Horizontal curves are paths in sub-Riemannian manifolds that are tangent to the horizontal distribution, representing the shortest paths accessible within a given geometric structure. These curves are significant because they provide insight into the intrinsic geometry of sub-Riemannian spaces, where the usual notion of distance can differ from traditional Riemannian manifolds due to the constraints imposed by the horizontal distribution.
Horizontal distribution: Horizontal distribution refers to a way of organizing and structuring the tangent spaces in a sub-Riemannian manifold, where the available directions for movement are limited to a specified subset of tangent vectors. This concept is crucial in understanding the geometry of sub-Riemannian manifolds, as it defines how paths can be traversed within the manifold and plays a key role in determining the manifold's geometric properties. By establishing horizontal distributions, one can analyze the behavior of curves and their lengths in these constrained settings.
Horizontal layer: A horizontal layer refers to a concept in geometric measure theory that involves the stratification of a space into layers of constant height or depth, particularly in the context of sub-Riemannian manifolds and Carnot groups. This structure is crucial for understanding the intrinsic geometry and analysis on these spaces, as it helps to define notions like volume, area, and various geometric properties that depend on how the layers interact with each other and the underlying metric. In particular, horizontal layers are closely related to the idea of curves and surfaces defined by horizontal directions given by the underlying distribution of the manifold or group.
Hörmander condition: The hörmander condition is a criterion that determines the controllability of systems described by differential equations, particularly in the context of sub-Riemannian geometry. It states that a distribution generated by a set of vector fields satisfies the condition if the Lie brackets of these vector fields span the tangent space at every point. This condition is crucial for analyzing both the geometric properties of sub-Riemannian manifolds and for applications in control theory and robotics, ensuring that paths can be effectively planned and controlled.
Image Processing: Image processing refers to the manipulation and analysis of digital images through various algorithms to enhance, extract, or analyze information. It is a crucial aspect of computer vision and plays a significant role in measuring geometric properties and understanding structures within images, making it relevant for understanding dimensions and shapes in mathematical contexts.
Jean-Pierre Serre: Jean-Pierre Serre is a renowned French mathematician known for his significant contributions to algebraic topology, algebraic geometry, and number theory. His work has paved the way for further advancements in the fields of geometry and analysis, influencing various areas including the study of sub-Riemannian manifolds and Carnot groups.
Lie Algebra: A Lie algebra is a mathematical structure that studies the algebraic properties of symmetry through the use of a vector space equipped with a binary operation known as the Lie bracket. This bracket operation captures the essence of the commutation relations between elements, leading to deep insights in geometry and physics. Lie algebras are crucial for understanding the behavior of continuous transformation groups, particularly in the context of sub-Riemannian manifolds and Carnot groups, where they help describe the intrinsic geometry and symmetries of these spaces.
Lie group: A Lie group is a mathematical structure that combines algebraic and geometric concepts, specifically a group that is also a smooth manifold. This means that the group operations of multiplication and inversion are smooth, allowing for the application of calculus. Lie groups play a crucial role in connecting symmetries and differentiable structures, particularly in areas like geometry, physics, and control theory.
Measure-theoretic properties: Measure-theoretic properties refer to the characteristics of sets and functions that can be analyzed using the framework of measure theory, which is a branch of mathematics dealing with the quantitative aspects of size, volume, and probability. These properties are fundamental for understanding how to assign measures to sets in a rigorous way and help bridge concepts between geometry and analysis. They are crucial for exploring continuity, convergence, and compactness in various mathematical structures.
Mitchell-Bellaïche Theorem: The Mitchell-Bellaïche Theorem is a result in geometric measure theory that provides insights into the structure of sub-Riemannian manifolds and their geodesics. It establishes conditions under which the Hausdorff measure of a subset of a sub-Riemannian manifold can be characterized in terms of the geometry of the underlying Carnot group. This theorem plays a crucial role in understanding how geodesics behave and interact in these complex geometrical structures.
Poincaré Inequality: The Poincaré Inequality is a fundamental result in analysis that provides a bound on the average deviation of a function from its mean value in terms of its gradient. It connects the concepts of function space, measure, and differential calculus, especially in the context of spaces with geometric structures like sub-Riemannian manifolds and Carnot groups. This inequality plays a crucial role in establishing the existence of certain properties like Sobolev embeddings and the analysis of differential equations in these specific geometric settings.
Robotic motion planning: Robotic motion planning refers to the process of determining a sequence of movements or actions that a robot must take to move from a starting point to a target location while avoiding obstacles and optimizing performance criteria. This involves considering the geometry of the environment, the dynamics of the robot, and constraints such as safety and efficiency. In the context of sub-Riemannian manifolds and Carnot groups, motion planning becomes particularly intricate due to the limitations in movement directions and the geometric structures involved.
Sobolev Spaces: Sobolev spaces are mathematical constructs that extend the concept of classical spaces of functions, incorporating both function values and their derivatives. They are essential in the study of partial differential equations, allowing for the analysis of weak derivatives and providing a framework for measuring the smoothness of functions in various contexts.
Step of a group: The step of a group refers to the minimal number of commutators needed to express any element of the group in terms of the generators. In the context of certain mathematical structures, particularly in the study of sub-Riemannian manifolds and Carnot groups, the concept of step is crucial for understanding the underlying geometric and algebraic properties. A group can be classified by its step, which helps in analyzing the complexity and structure of the group, particularly how elements can be generated from one another.
Stratification: Stratification refers to the process of decomposing a space into a finite collection of smooth manifolds that are organized in a specific manner, allowing for analysis and understanding of geometric structures. In contexts such as sub-Riemannian manifolds and Carnot groups, stratification aids in identifying the layers or levels of differentiability and geometry, which can greatly influence the behavior of curves and shapes within these spaces.
Sub-Laplacian: The sub-Laplacian is a second-order differential operator associated with sub-Riemannian geometry, generalizing the Laplacian operator to settings where the usual smoothness and dimensionality conditions may not hold. It plays a crucial role in defining notions of harmonicity and analysis on sub-Riemannian manifolds, which have a distinct structure from classical Riemannian manifolds. This operator is essential for studying the behavior of functions within Carnot groups, particularly in understanding geometric properties and the underlying structures.
Sub-Riemannian manifold: A sub-Riemannian manifold is a type of geometric structure that generalizes the concept of Riemannian manifolds by introducing a distribution of tangent spaces that allows for the definition of paths constrained to a specific subset of directions. This structure is important in studying Carnot groups, which are examples of sub-Riemannian manifolds, as they focus on the intrinsic geometry and analysis defined by these constraints. Additionally, understanding Hausdorff dimension and measure in these spaces is crucial, as it provides insights into how geometric properties behave under the imposed directional limitations.